BSTJ 54: 6. July-August 1975: A Method for Calculating Rain Attenuation Distributions on Microwave Paths. (Lin, S.H.)

Copyright © 1975 American Telephone and Telegraph Company

The Hell System Technical Journal

Vol. 54, No. 6. July-August 1975

Printed in U.S.A.

A Method for Calculating Rain Attenuation
Distributions on Microwave Paths

By S. H. LIN

(Manuscript received January 9, 1975)

An engineering method is proposed for calculating rain attenuation
distributions for frequencies greater than 10 GHz and for paths of arbitrary
length. The technique is based upon the observed approximate lognormality
of rain attenuation and rain rate statistics ivithin the range of interest;
it reflects local meteorology through incorporation of the observed point
rain rate distribution. Some important parameters in the resulting formulas
are determined empirically from experimental data. Sample calculated
results agree well with available experimental data from Georgia, New
Jersey, and Massachusetts. This new technique may prove useful for
engineering radio paths at frequencies above 10 GHz. Sample calculations
of expected outage probability are given for 11- and 18-GHz radio links
at Atlanta, Georgia, as a function of repeater spacing and transmission
polarization .

I. INTRODUCTION

An important problem in designing radio relay systems at fre-
quencies above 10 GHz is the radio outage caused by rain attenuation.
Determination of the appropriate radio repeater spacings for economic
and reliable operation requires a knowledge of the probability dis-
tribution of rain attenuation as a function of repeater spacing at
various geographic locations. Most available data on rain rate sta-
tistics are measured by a single rain gauge at a given geographic loca-
tion. A procedure for calculating a rain attenuation distribution from
a point rain rate distribution is, therefore, needed.

The results of rain gauge network measurements 1-5 indicate, how-
ever, that the measured short-term distributions of point rain rate
vary significantly from gauge to gauge. For example, at Holmdel,
New Jersey, there was considerable variation 1 among the measured
point rain rate distributions obtained from 96 rain gauges located in
a grid with 1.3-km spacing over a 6-month period. Among these 96
distributions, the incidence of 100-mm/h rains is higher by a factor of

1051

5 for the upper quartile gauges than for the lowest quartile (see Fig.
32, Ref. 1). Data from a rain gauge network in England 5 indicate
that even with a 4-year time base and averaging over observations by
four gauges with 1-km gauge spacing, the four-gauge-average rain
rate incidence can differ by a factor of 3 for rain rates above 80 mm/h,
depending on which four gauges are chosen for averaging. This means
that, on a short-term basis, the relationship between the path rain
attenuation distribution and the rain rate distribution measured by a
single rain gauge is not unique. The prediction of a path rain attenua-
tion distribution from a point rain rate distribution is, therefore,
meaningful only if the time base is sufficiently long to yield stable,
representative statistics. Accordingly, knowledge of the long-term
statistical behavior of point rain rate and path rain attenuation is
essential for radio path design.

The available experimental rain data (Appendix B, Figs. 10 to 13,
and Ref. 6) indicate that the long-term distributions of point rain
rate R and rain attenuation a are approximately lognormal within
the range of interest to designers of radio paths using frequencies
above 10 GHz. This paper describes a method for calculating rain
attenuation distributions based upon this lognormal hypothesis.

A lognormal distribution is uniquely determined by three param-
eters (see Section 2.1). A set of equations are derived to relate the
lognormal parameters of attenuation a to those of point rain rate R.
Thus, given a long-term, representative distribution of point rain rate
at a given geographic location, the rain attenuation distribution for
any path length of interest can be calculated. The method is outlined
as follows.

The available theory 7-13 for converting rain rate R into rain attenua-
tion gradient /3 in dB/km is appropriate to spatially uniform rain
rates, whereas actual rainfalls are usually not uniform over an entire
radio path. To apply the uniform rain theory, the radio path volume
is divided into small incremental volumes AV, in which the rain rate
is approximately uniform. The rain rate R in each small volume A V is
associated with a corresponding attenuation gradient /3 by the uniform
rain theory. The total path attenuation a is then the integral of /3 over
the path volume.

If the spatial distribution of the attenuation gradient /3 were uniform,
the path attenuation at a given probability level would increase
linearly with path length. On the other hand, if the spatial distribution
of /3s were not uniform and the time fluctuations of the /3s were sta-
tistically independent, the incremental attenuation contributed by
each AV would sum on an rms basis. Intuitively, we feel that the
attenuation gradients at two different positions are partially corre-

1052 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

lated, with a correlation coefficient that is a decreasing function of
the spacing between the two positions. This behavior is described by
introducing a spatial correlation function for /3. This spatial correlation
function is used in the calculation of lognormal parameters of path
attenuation a from those of attenuation gradient 0.

In this formulation, the appropriate incremental sampling volume
AV is of the order of 1 m 3 and the corresponding appropriate rain
gauge integration time about 2 s, requiring, therefore, 2-s point rain
rate distributions (see Section 2.3).

Tables II and III and Figs. 5 to 8, discussed in Section IV, present
comparisons of calculated and measured attenuation distributions.
The satisfactory correspondence appears to validate the method of
calculation. Section V and Fig. 9 present the calculated results for the
outage probabilities of 11- and 18-GHz radio links in Georgia. Section
VI discusses some qualifications to the methodology.

Supporting material and mathematical derivations are given in
Appendices A to D. Appendix E lists symbols and their definitions.

II. BASIC DEFINITIONS AND FORMULATION

2.1 Lognormal distributions of attenuation and rain rate

The equations approximating the rain attenuation distribution and
point rain rate distribution are :

P[a(L) ^ A] c* Po(L) -ferfc [ ln A ^* m ] (D

and

P(R ^ r) ~ Po(0) • fterfc [ ln '" "J^" 1 ] , (2)

where erfc (~) denotes the complementary error function, ln (~)
denotes natural logarithm, *S„ and Sr are the standard deviations of
lna and ln R, respectively, during the raining time, a m and R m are the
median values of a and R, respectively, during the raining time, Po(L)
is the probability that rain will fall on the radio path of length L, and
P (0) is the probability that rain will fall at the point where the rain
rate R is measured. The definition and the determination of raining
time, and hence Pa{L) and Po(0), are discussed in Section 2.5. The
measured distribution of point rain rate is a function of rain gauge
integration time 7 1 . 14-20 The appropriate integration time is about 2 s
in our formulation as discussed in Section 2.3.

2.2 Radio path definition

C. L. Ruthroff 21 has defined a "radio path," giving a physical as
well as mathematical representation in the spatial volume significant

RAIN ATTENUATION 1053

to propagation considerations (Fig. 1). In essence, the "radio path"
corresponds to the first Fresnel zone, a prolate ellipsoid of revolution
terminated at the ends by the transmitting and receiving antennas.

Since the first Fresnel zone is circularly symmetric with respect to
the path axis connecting the transmitting and receiving antennas, we
adopt a cylindrical coordinate system (Fig. 1) coaxial with the path axis
(z-axis) in the formulation.

The radius h(z) and the circular cross section Q(z) (see Fig. 1) of the
radio beam at a distance z from the transmitter are

»„_[»*<*=•>]• (3)

and

Q(z) = Th*(z), (4)

where X is the radio wavelength and L is the distance between trans-
mitter and receiver. For example, at 18 GHz on a 5-km path, the
average beam radius, the average beam cross section, and the radio
path volume are about 3 m, 30 m 2 , and 150,000 m 3 , respectively.

2.3 Path integral formulation for rain attenuation

The spatial distribution of actual rainfall is usually nonuniform.
The rain density, the point rain rate R, and the corresponding (point)
rain attenuation gradient are all functions of position (p, <p, z) and of
time (t). The total rain attenuation a in dB incurred on a radio path of
length L (Fig. 1) is calculated by integrating the incremental attenua-
tion da along the path

«(0 = f L ^T dz = [ L P q (.z,t)dz (5)

Jo dz Jo

rh 1 /-2t rh(z)

= / 7171 / / Mp> *i z > Opdpd^dz, (6)

Jo lJ{z) y^.=o J P =o

where

1 /-2r rh(z)

P 9 (z, = n77\ / / Mp> *' 2 ' i)pdpd<t> (7)

y{z) j $ = o J p-o

is the average value of /3(p, 0, z, t) over the radio beam cross section
Q(z) at a distance z from the transmitter, and

da{z, t) = q (z, t)dz (8)

is the incremental attenuation experienced in the incremental segment
dz at a distance z from the transmitter.

To shorten the notations in the following equations, we use the
vector s to denote the position (p, 0, z) and dv to denote pdpdfdz in the
volume integration.

1054 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

h|Z):RADIUS OF RADIO BEAM CROSS SECTION AT Z

Fig. 1— Configuration of a radio path.

The rain density and the rain attenuation gradient /3 in dB/km are
only meaningful with respect to a sampling volume AV large enough
to contain sufficient raindrops to yield stable volume average quanti-
ties. The results of measurements of rain density and rain rate by a
photographic method 22 indicate that the typical rain density at 1-mm/h
rain rate varies from 50 to 100 raindrops per m 3 , depending on geo-
graphic location. This means, on the average, a 0.1-m 3 volume con-
tains only 2 raindrops at 0.25-mm/h rain rate. Such a small sampling
volume will not measure the rain rate in the conventional sense, but
rather will "see" individual raindrops. Thus, for a meaningful mea-
surement of rain rate below 1 mm/h, the sampling volume should be
at least 1 m 3 .

On the other hand, the available theoretical results 7-13 relating
and R assume that the rain density is uniform within the volume
of interest. To use the available functional relationship P(R) to
convert the statistics of R into that of /3, the sampling volume AV
must be sufficiently small so that the rain density (and the rain rate)
is approximately uniform within AV. The observations by a capacitor
flow rain gauge 2324 and by a raindrop photographic method 25 indicate
that heavy rain has fine scale structure on the order of 1 m. This
means AV should not be much larger than 1 m 3 .

These two constraints indicate that the sampling volume should be
on the order of 1 m 3 . We have chosen*

AV

1 m 3

(9)

in our formulation.

* A choice of AV somewhat different from 1 m 3 is also possible. Since the spatial
correlation coefficient of /3 depends on the sampling volume, the use of a slightly
different AV would result in a slightly different characteristic distance G, defined in
eq. (35) and determined in Section 4.1. For example, a larger sampling volume, with
more smoothing effect, will result in a larger characteristic distance G for /3.

RAIN ATTENUATION 1055

Therefore, in this paper, 0(s, t) is defined as the rain attenuation
gradient at time t, owing to rain in a 1-m 3 sampling volume, AV,
centered at s, and /3,(z, t) is the average value of /3(s, t) over the path
cross section Q(z).

If A is the area of the collecting aperture of a rain gauge and V R
the average descent velocity of rainfall, then the appropriate rain
gauge integration time T±v to measure rain rate in a 1-m 3 sampling
volume AV is

r — arfc- (10)

For example, if A„ = 0.073 m 2 (i.e., 12-in. diameter), then T± v is
about 2 s, assuming V K =7 m/s. In the measurements of raindrop size
distributions, Laws and Parsons 26 have also used an integration time
in the order of seconds during heavy rain. The Laws-and-Parsons rain-
drop size distribution is the basis of most uniform rain theories for
converting R into 0. We therefore define R(s, t) as the point rain rate
measured by a rain gauge with integration time T&v, located at s. The
shape of the 1-m 3 sampling volume denned by the rain gauge is cy-
lindrical and is considerably different from that of the incremental
AV. We assume that the long-term distributions of rain rates for these
two different shapes of 1-m 3 sampling volume are approximately the
same.

Based upon these definitions, we postulate that the long-term prob-
ability distribution of R(s, t) can be converted into the long-term
probability distribution of 0(s, t) by a relationship discussed in the
next section.

The integration times of most available point rain rate data are
longer than the T^v (=2 s) required by this formulation. The depen-
dence of point rain rate distribution on the rain gauge integration time
in the range

1.5b £ T £ 120s (11)

has been determined by Bodtmann and Ruthroff" for a 2-year (1971-
1972) measurement at Holmdel, New Jersey. By using this experi-
mental result and interpolation, we convert the available point rain
rate distribution with T in the range (11) into a 2-s point rain rate
distribution.*

2.4 Average relationship between rain rate and rain attenuation gradient

The instantaneous relationship obtaining between the point rain
rate R(s, t) and the corresponding rain attenuation gradient |8(s, t)

* This relationship has not yet been demonstrated to be geographically independent.
1056 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

100 -

I io

0.1

Y,„ H V h V H

• CALCULATED FOR V-POLARIZATION
O CALCULATED FOR H-POLARIZATION
APPROXIMATION 8Y = 7 • R^

-FROM REF.7

■ ■ I

10"

10

10

40

RAIN ATTENUATION GRADIENT IN dB/Km

Fig. 2 — Theoretical relationship between rain rate and rain attenuation gradient.

depends upon the particular distribution of raindrop sizes, shapes, and
orientations, the speed and local direction of the wind, and the rain
temperature. The average relationship, assuming uniform rain density,
spherical raindrops, and Laws-and-Parsons drop-size distribution, has
been calculated by Ryde and Ryde, 11-13 Medhurst, 9 and Setzer. 10
Recently, Morrison, Cross, and Chu, 7 - 27 and Oguchi 8 have refined
these calculations by including the effects of nonspherical raindrops.
Figure 2 shows this theoretical relationship* for transmission frequen-
cies of 11, 18, and 30 GHz.

Many authors have pointed out that this average relationship be-
tween the rain rate R and rain attenuation gradient /3 can be approxi-
mately described by

= y(\)-R*™, (12)

where 7(X) and 77 (X) depend upon the radio wavelength X and the
polarization of the radio signal. Table I lists the estimated y(X) and
77 (X) for 11, 18, 30, and 60 GHz.

* In Fig. 2, the average of the absolute value of raindrop canting angle is assumed to
be 25 degrees. This angle has been found representative by Chu (Refs. 28 and 29) in
comparisons of calculated results with experimental observations (Refs. 30 and 31 ) of
the differential rain attenuation experienced by horizontally and vertically polarized
signals on the same radio path.

RAIN ATTENUATION 1057

Table I — Parameters relating rain rate R and rain attenuation

gradient p
p = yRn p in dB/km R in mm/hour

Frequency

7

V

(GHz)

V-Pol

H-Pol

V-Pol

H-Pol

11

18.1
30
HO*

0.013
0.05
0.15
0.7*

0.015

0.054

0.17

0.7*

1.22
1.11
1.04
0.814*

1.23
1.14
1.04
0.814*

* The 60-CiHz parameters are estimated from results in Ref. 10 in which only
spherical raindrops are considered.

Taking logarithms of both sides of eq. (12) yields
ln/3 = In 7 + Tj-ln R.

(13)

From this equation, if the distribution of point rain rate R is approxi-
mately lognormal in the range of interest, then the distribution of
attenuation gradient will also be approximately lognormal (see
Appendix B). The distribution of can therefore be written as:

P(^fi)^f.(0)ierfc[ ' nJ? vz :^ g '

]•

(14)

where p m is the median value of during the raining time and Sp is
the standard deviation of In R during the raining time. Furthermore,
eqs. (12) and (13) imply the relationships

S p = vSr (15)

and

p m = yRl. (16)

Equations (15) and (16) allow us to convert the lognormal distribu-
tion (2) of R into the lognormal distribution (14) of R, and vice versa.

2.5 Rainfall probability P„(0) and raining time

In principle, the probability of raining, Po(0), is obtained as the limit

lim P(R ^ e) = P (0). (17)

«-»o+

An instant t is considered to be raining time if the condition

lim R{t) > e

(18)

is satisfied. The lower cutoff threshold in most presently available rain

1058 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

S 1 5-

5 3 -

RAIN GAUGE NETWORK DATA OF
FLORIDA THUNDERSTORM PROJECT
(REF.20)

-

°^ EMPIRICAL CURVE DESCRIBED
BY EQ. (20)

-

S'

-

o/

-

ORLANDO.FLORIDA
2-MINUTE INTEGRATION TIME

-

i l l 1 1 _l 1 1 1 L_

1 1 l 1 1 1 1 1 1 1 1

70

PATH LENGTH L IN KM

Fig. 3 — Rain-gauge network data on path length dependence of rainfall proba-
bility P a (L).

rate data is about 0.25 mm/h. Therefore, in practice, we approximate
0+ in definitions (17) and (18) by 0.25 mm/h. The rationale for this
approximation is twofold.

(i) Rain rates below 0.25 mm/h have practically no significant
effects on radio communication links at frequencies below 60

GHz.
(ii) Rain rates below 0.25 mm/h cannot be measured accurately
by most existing rain gauges with standard recording strip
charts.

At the present time, the probability P(R ^ 0.25 mm/h IT = 1 min)
is available at only a few locations. 1418 - 20 For most locations, we can
obtain P(R ^ 0.25 mm/h) with 1-h integration time from the Weather
Bureau hourly precipitation data. 32 The experimental results on the
effect of rain gauge integration time T on P„(0) in Florida 1420 and
Japan 18 indicate that

P(R ^ 0.25 mm/h | T g 1 min)

~ 0.5- P(R ^ 0.25 mm/h|T = 1 h). (19)

Therefore, we use Weather Bureau data and approximation (19) to
estimate Po(0) at several locations of interest where direct measure-
ment of Po(0) with 1-min integration is not available.

RAIN ATTENUATION 1059

Intuitively, we expect the probability Po(L) of rainfall on a radio
path of length L to increase with L, since a longer path has a higher
chance of intercepting rain of limited extent. From the rain gauge
network data (Fig. 3) of the Florida Thunderstorm Project, 14 we
obtain the empirical formula

p„(D at i - r 1 - %%„ , (20)

[ 1 + 2IT 5 J

where L is in units of kilometers and

Po(0) = lim P (L) (21)

is the point rain probability that depends on geographic location. A
theoretical consideration leading to the empirical form (20) is dis-
cussed in Appendix C.

III. OUTLINE FOR CALCULATING RAIN ATTENUATION DISTRIBUTION

3.1 Path length dependence of median attenuation a m and standard
deviation S,,

From the lognormal approximations (1) and (14), it can be shown 33
that:

^ = In 1 1 + ^ J (22)

fll-Injl + d} (23)

where

a m = 5-ex P r-^] (24)

<3 m = ,3-ex P [^], (25)

a = E L [a(t)\ (26)

0=E o {P(t)} (27)

trl - E L {<*(t)} -a 2 (28)

and

4 = Eo{P(t)} -f. (29)

El\~) denotes a statistical (time) average under the condition that
rain is falling on the radio path of length L, £ {~} denotes statistical
(time) average under the condition that rain is falling at the location

1060 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

of interest. In eqs. (27) and (29), we assume that the long-term, large-
sample, conditional statistical average and variance a} are inde-
pendent of position s in or near the radio path of interest. Therefore,
we omit the position s argument in these equations.

By using eqs. (6), (22), (23), (24), and (25), we can derive formulas
for the dependence of S a (L) and a m (L) on the radio path length L. The
lengthy derivations are given in Appendix A. The results are

tf(L) = lnPo(L) jl + H(L) [^^ -l]} (30)

and

-< i >-'-- i -£$h ,p [ a T^]' (31)

where

w - b f I!w)L fw-)L J*- is - s ' )dvd °' (32)

*„(s f s') = 4- [EuWb, t) -0(8', 0] - A) (33)

is the spatial correlation coefficient 34 ' 35 between 0(s, t) and 0(s', t), and

C P (S,S') = c 2 pu-rf, u (s,s') (34)

is the spatial covariance function 34 ' 35 of /3(s, t) and 0(s',t), E u {~]
denotes the unconditional statistical (time) average including both
raining time and nonraining time, /3 U and o% u are the unconditional
statistical mean and variance, respectively, of /3 as defined by eqs.
(48) and (49) in Appendix A.

In eq. (32), the integration volume is the entire radio path (Fig. 1)
and is a function of both path length L and wavelength X. However,
the spatial correlation coefficient \f/ u of (point) rain attenuation
gradient denned in eq. (33) is not a function of radio path length L.

If the random fluctuations of the 0s were "coherent" along the
entire radio path, then $ u , H(L) and P (L)/P (0) would be identi-
cally unity. Under such conditions, S a would be identical to Sp
and a m (L) would be equal to m -L as expected. The complexity
of path length dependencies of S a and a m in eqs. (30) to (34) is caused
by the partially correlated, random fluctuations of 0s at various points
in the radio path.

We postulate \f/ u to have the functional dependence on distance

g(X, AV) m)

*" = [G 2 (X, AV) + #:» l }

RAIN ATTENUATION 1061

within the range of interest, where

d = |s — s'|

is the distance between the two observation points (s) and (s'), and
G(\, AV) is a characteristic distance at which ip u = l/v2. The de-
pendence of /3 on wavelength X (Fig. 2) and sampling volume A V
(Section 2.3) indicates that the characteristic distance G may also be
a function of X and AV. However, in Table I, the exponents i\ are all
very close to unity for frequencies ranging from 11 to 60 GHz. This
means /3 is approximately linearly proportional to R in this frequency
range. Therefore, the characteristic distance G of /? is approximately
equal to that of point rain rate R and will not be very sensitive to fre-
quency in the range from 11 to 60 GHz.

Substituting (35) into (32) and carrying out integrations over p'
and 4>' yield

H{L) --»-h dB Lw)h pdp hw)> (36)

where

F = VPT — Vf + pcos^-ln pr^

Vf — p cos

(37)

TF = G 2 + (2 — z') 2 + p 2 + h 2 (z') - 2 P h(z') cos (38)

f = G 2 + p 2 + (2 - z') 2 (39)

= 4>-4>'. ( 4 °)

The remaining integrations can be carried out numerically by com-
puter. The calculated H(L) for G = 0.75, 1.5, and 3 km, respectively,
are shown in Fig. 4.

Notice that the radius h(z) of a "radio beam cross section" is on the
order of several meters, whereas the characteristic distance G is on the
order of kilometers (see Section 4.1). Therefore,

G»h(z) (41)

for most radio paths at frequencies above 10 GHz. Imposing the con-
dition (41) reduces the complicated integrations in eq. (36) to the
simple result

1G 1

H§ + Jl + gl-Jl + g + l

(42)

The differences in numerical values of H(L) calculated by (36) and
by approximation (42) are less than 0.1 percent within the range of
interest. Notice that H(L) is practically independent of wavelength

1062 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

0.8

0.6

G=0.75

15 20 25

PATH LENGTH L IN KM

35

40

Fig. 4 — Dependence of H(L) on path length L and characteristic distance G.

X because of condition (41). Therefore, we omit the wavelength speci-
fication on Fig. 4.

Thus, we have obtained all the necessary relationships. The pro-
cedure for calculating rain attenuation distributions from point rain
rate distributions is summarized in the next section.

3.2 Procedure for calculating rain attenuation distribution

(i) Convert the measured distribution of point rain rate with
T ^ 2 min into the distribution of 2-s point rain rates by the
conversion factor in Ref. 15.
(ii) If Po(0) with T ^ 1 min is not available at the location of
interest, use approximation (19) and the Weather Bureau
hourly precipitation data to estimate Po-
(Hi) Estimate the lognormal parameters R m and Sr of the 2-s
point rain rate distribution by a least-squares approximation.
This step is carried out by a computer iteration process to
obtain the (R m , Sr) pair that minimizes the differences (i.e.,
the sum of squares of errors) between the data points and the
lognormal approximation.
(iv) Calculate m and S by formulas (15) and (16).
(») Calculate P (L), a m (L), and S a (L) by formulas (20), (30), and

(31).
(vi) Substitute Po(L), a m (L), S a (L) into eq. (1) to give the attenua-
tion distribution.

RAIN ATTENUATION 1063

IV. COMPARISON OF CALCULATED RESULTS WITH EXPERIMENTAL DATA

The measured rain attenuations in many experiments contain not
only the path rain attenuation but also the transmission loss owing to
wet radomes. The presently available information is insufficient for
accurate estimation of wet radome attenuation as a function of rain
rate, wind direction, radome shape, size, material, and surface aging
effects.

Based upon two measurements of wet radome attenuations dis-
cussed in Appendix D, we assume that a flat, vertical radome causes
1.5-dB attenuation during heavy rain. Therefore, 3-dB attenuation,
caused by a pair of wet radomes, is added to the calculated path rain
attenuation and the result is compared with the measured data
utilizing such radomes. In some experiments, the flat radomes are
slanted inward to further reduce wetting the radome surfaces. The
attenuation caused by a pair of such radomes during rain is assumed*
to be less than 3 dB. More detailed discussion of the radome problem
is given in Appendix D.

4.1 Determination ot characteristic distance G

From the 3-year (1971-1973) distribution of 1-min point rain rates
measured at Merrimack Valley, Massachusetts, 36 the conversion factor
in Ref. 15, and the Weather Bureau data, 32 we obtain the following
approximate lognormal parameters of the distribution of 2-s point rain
rate:

P„(0) ~ 3.3%, (43)

R m S* 1.23 mm/h, (44)

and

S R ~ 1.34. (45)

Following the procedure outlined in Section 3.2, we use these param-
eters to calculate a family of rain attenuation distributions as a func-
tion of the distance parameter G. Figure 5 displays the results for an
18-GHz, 4.3-km path subject to Merrimack Valley rain and makes
comparison with measured data (1971-1973) at the same location
(Ref. 36 and Tables II and III). The radomes on this path are vertical
and almost flat. The solid curves on Fig. 5 are calculated path rain
attenuations plus assumed 3-dB radome attenuation. Figure 5 indi-
cates that

G S 1.5 km (46)

provides good agreement; therefore, a 1.5-km characteristic distance 39,40

* The slanted radomes may get wet during some heavy rains accompanied by
strong wind gusts.

1064 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

10

<

•A 10- 2

10"

10"

MERRIMACK VALLEY.IVWSS.

18GHz.4.3 km, V-POL

3 YEARS (1971-1973)

CALCULATED PATH ATTENUATION +3 dBWET

RADOME ATTENUATION

G=0.75

20 30 40

ATTENUATION IN DECIBELS

Fig. 5 — Determination of characteristic distance G by comparing experimental
data with calculated results (solid lines) using rain rate data in Fig. 10.

is used for the calculations and comparisons with other sets of data at
other locations in the following sections.

From eqs. (35) and (46), it is easily shown that d ^ 15 km for a
spatial correlation coefficient, \f/ u ^ 0.1. In other words, a "rain cell,"
based upon a definition of ^„ ^ 0.1 within the cell, has a typical
spatial extent of 15 km. Obviously, the cell size depends on its definition.

4.2 Comparisons of calculated results with data in Georgia

Figure 6 compares the calculated result with data from a 5.1-km
17.7-GHz path at Palmetto, Georgia, measured during two 1-year
periods (November 1970 through October 1971 and August 1973
through July 1974).' The radomes on this path are flat and canted
inward. The calculated result is based upon the rain rate distribution
measured by a tipping-bucket rain gauge at Palmetto in the same time

'The data from November 1971 to July 1973 are not used because of inter-
mittent troubles in the rain gauge and the magnetic tape recorder.

RAIN ATTENUATION 1065

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1066 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

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RAIN ATTENUATION 1067

1

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RICO-PALMETTO.GA.

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17.7 GHz,5.1 KM. H-POL

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— — — CALCULATED PATH ATTENUATION +3 dB

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ATTENUATION IN DECIBELS

Fig. 6 — Comparison of 17.7-GHz rain attenuation data at Palmetto, Georgia, with
calculated result (solid line) using rain rate data in Fig. 12.

period. Figure 6 has two calculated attenuation curves, one without
radome attenuation and another with 3-dB radome attenuation.

In the 11-GHz band, the path lengths of interest may be as long as
50 km. It is desirable to test the validity of the method for long paths.
A preliminary comparison is shown in Fig. 7 for an 11-GHz, 42-km
path between Atlanta and Palmetto, Georgia. The radomes on this
path are flat and almost vertical. The attenuation and rain rate were
observed during a 1-year period (August 1973 through July 1974).
The calculated-plus-3-dB-radome-loss result is reasonably close to the
data. In Figs. 7 and 13, notice that the measured attenuation and
rain rate distributions are both somewhat higher than the lognormal
approximations in the probability range from 10 -2 to 4 X 10~ 2 percent
time. We believe that these deviations are an artifact of the short
observation time. A more critical test of this method for long paths
awaits longer-term data.

1068 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

4.3 Comparison of calculated result with data in New Jersey

Rain attenuation experiments on two paths (18 GHz, 6.4 km, and
30.9 GHz, 1.9 km) were carried out simultaneously in 1968 and 1969
at Holmdel, New Jersey. 38 However, local point rain rate distributions
were not measured during this period. On the other hand, Bodtmann
and Ruthroff 15 have suggested a method for relating point rain rate
distribution to path rain attenuation distributions on short paths.
Thus, by using the short-path attenuation data from the 30.9-GHz,
1.9-km path and Bodtmann and RuthrofFs method, the 2-year dis-
tribution of point rain rate at Holmdel was estimated as shown in
Fig. 11. Based on this estimated rain rate distribution, we calculate
the 2-year rain attenuation distribution on an 18-GHz, 6.4-km path
and compare this with the measured result in Fig. 8. The radomes in
these experiments are flat, slanted inward, and shrouded by substantial
wooden rain shields. We believe that the wet radome attenuation, with

s io-

<

| io-

ATLANTA-PALMETTO.GA.
11.6 GHz.42 KM.V-POL.
1 YEAR (8/73-7/74)

CALCULATED PATH ATTENUATION +3 dB WET
RADOME ATTENUATION

10

20 30 40

ATTENUATION IN DECIBELS

Pig 7 — Preliminary comparison of 1-year rain attenuation data from an 11-GHz,
42-km path in Georgia with calculated results (solid line) using rain rate data in
Fig. 13.

RAIN ATTENUATION 1069

1

r 1

I HOLMDEL.N.J.

18.5 GHz.6.4 KM.V-POL

2 YEARS (1968.1969)

<

CO

CALCULATED PATH ATTENUATION ONLY

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ATTENUATION EXCEEDS

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ATTENUATION IN DECIBELS

bO

Fig. 8 — Comparison of 18.5-GHz rain attenuation data in Holmdel, New Jersey,
with calculated result (solid line) using rain rate data in Fig. 11.

such protection, is negligible. Therefore, the calculated curve in Fig. 8
contains only path rain attenuation.

Figures 6, 7, and 8 show that the calculated results agree reasonably
well with measured data.

4.4 Excluded rain attenuation data

Many sets of rain attenuation data in the literature are not included
in the comparisons in Sections 4.2 and 4.3 because of one or more of
the following reasons.

(i) Many experiments used cone-shaped or hemispheric-shaped
radomes. The transmission loss resulting from rain running on
a pair of such radomes may vary from to 14 dB depending
on frequency, rain rate, and radome surface aging. The un-
certainty in estimating such radome attenuation is too large
for a meaningful comparison between the data and the cal-
culated path rain attenuation distribution.

1070 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

(it) The antennas in the experiment are not covered by any
radomes, exposing the antenna feeds and reflecting surfaces
to rain, snow, and ice. No information is available for esti-
mating the possible transmission loss from the wetting of these
elements.

(in) The published information does not specify whether the
antennas are covered by radomes or not. The configuration
of the radome, if used, is also unknown.

(iv) The polarization of the transmitted signal is unstated.
(v) The time base of the experiment is too short to yield long-term
representative statistics.

(vi) No rain rate data with T ^ 2 min is available at or near the
location of the rain attenuation experiment.

V. OUTAGE ESTIMATION FOR 11- AND 18-GHz RADIO LINKS

For a constant transmitter output power, the dependence of fade
margin F (L) in dB on the path length L is

F (L) = F Q (L ) - 20 logio ( ^ ) dB, (47)

where L is a reference repeater spacing and F (L ) is the corresponding
reference fade margin. For 11- and 18-GHz radio, reasonable clear-day

reference fade margins are

Fo = 40 dB for 18 GHz at L = 4 km
and

Fo = 40 dB for 11 GHz at L„ = 40 km.

A radio outage occurs when the path rain attenuation plus the wet
radome attenuation exceeds the clear-day fade margin F . By sub-
stituting the fade margin (47) into the attenuation distribution (1), we
can calculate the probability of radio outage per hop as a function of
repeater spacing L. As an example, Fig. 9 shows the outage probabili-
ties' for 11- and 18-GHz radio links in Atlanta, Georgia.

The wet radome attenuation A « is assumed to be 3 dB in these cal-
culations of outage probabilities.

VI. SOME QUALIFICATIONS

This section discusses some limitations, approximations, and assump-
tions in the theoretical calculation procedure, the data employed, and
the calculated results.

* Multipath interference fading can also cause outages. An empirical formula for
estimating the multipath-caused outage probability can be found in Ref. 41.

RAIN ATTENUATION 1071

1000

100

5 10

18 GHz H-POLARIZATION

11 GHz V-POLARIZATION

18 GHz F o (4KM)=40dB
11 GHz F o (40KM)=40 dB
RADOME ATTENUATION =3 dB

15 20

HOP LENGTH IN KM

25

35

Fig. 9 — Expected outage times of 11- and 18-GHz radio links as a function of hop
length in Atlanta, Georgia.

6.7 Uncertainty in estimation of lognormal parameters P„, R,„, and Sn

Some point rain rate measurements report only the heavy rain (e.g.,
^ 30 mm/h) portion of the distribution, neglecting the light rain sta-
tistics completely. Table III indicates that the median rain rates R m
at many locations are less than 4 mm/h. In other words, the major
portion (=98 percent) of the distribution is missing, and accurate
estimation of the statistical parameters R m and Sr from the tail
region (^2 percent) is difficult.

Furthermore, high rain rates (e.g., >140 mm/h) require a long
observation time to yield representative, long-term statistics. The
time bases of most available data may not be sufficient to yield stable
statistics for these extreme rain rates. For example, at Newark, New
Jersey, the 1-min point rain rate exceeded 180 mm/h only once in the
5-year period processed by Bodtmann and Ruthroff. To obtain rea-
sonably stable statistics, we need a sample size much larger than 1.
The omission of light-rain statistics together with the inherent in-
stability of the extreme rain rate statistics causes considerable un-

1072 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

certainty in the estimation of P , R m , and Sr. This uncertainty can
be reduced significantly if the light rain portion of the distribution is
also measured and reported.

6.2 Path length dependence of P

The empirical formula (20) for the dependence of Po on path length
L is obtained from the rain-gauge network data in Florida. The test
of the applicability of this empirical formula to other locations and
the improvement of this approximation will require further multiple
rain gauge experiments at other locations.

6.3 Dependence of point rain rate distribution on rain gauge integration time

The dependence of the point rain rate distribution on rain-gauge
integration time T has been obtained by Bodtmann and Ruthroff 15
from a 2-year experiment at Holmdel, New Jersey. Since a 2-year time
base may not be sufficient to yield stable statistics for high rain rates,
a longer time base may be needed to improve this empirical conversion
factor. The applicability of this result to other geographic locations
also remains to be verified.

6.4 Radome problem

In the comparisons of calculated and measured rain attenuation
distributions, the wet radome attenuations are assumed values. To
improve this approximation, a more systematic experimental study on
the dependence of radome attenuation on rain rate is needed.

6.5 Anisotropic spatial correlation f

At some geographic locations, the squall lines of heavy rain may
have a predominant orientation related to the predominant orienta-
tion of weather fronts. 42-44 This means the spatial correlation yp may
depend not only on the spacing but also on the orientation. However,
the presently available information is not sufficient for a quantitative
description of such an anisotropic correlation. Therefore, we use the
isotropic correlation coefficient (35) throughout our theoretical cal-
culations. Some of the difference between calculated and measured
attenuation distributions may be caused by neglecting the anisotropy
of the spatial correlation function.

VII. CONCLUSION

By using lognormal approximations, we have described a method
for calculating rain attenuation distributions on microwave paths.
The calculated results agree reasonably well with experimental data
in Massachusetts, New Jersey, and Georgia. This procedure may

RAIN ATTENUATION 1073

prove useful for the design of radio paths using frequencies above 10
GHz. To demonstrate the application, Fig. 9 shows the calculated
outage probability as a function of repeater spacing for 11- and 18-GHz
radio links in Georgia.

VIII. ACKNOWLEDGMENTS

I wish to express appreciation to K. A. Jarett who has helped in
developing a computer program for the numerical calculations in this
paper; to N. Levine, E. E. Muller, W. T. Barnett, D. C. Hogg, A. A. M.
Saleh, and T. S. Chu for valuable comments and suggestions that
greatly improved the consistency and precision of this analysis; to
D. C. Hogg, R. A. Semplak, R. A. Desmond, A. E. Freeny, and J. D.
Gabbe for rain rate and attenuation data at Holmdel; to W. T.
Barnett, M. V. Pursley, and H. J. Bergmann for attenuation data
in Georgia; to G. H. Lentz and J. J. Kenny for rain rate and attenua-
tion data in Massachusetts; and to C. L. Ruthroff, W. F. Bodtmann,
and A. L. Sims for rain rate data at many locations.

APPENDIX A

Derivation of Formulas Relating Rain Attenuation Distribution to Path Length

Since the random fluctuations of the attenuation gradient /3(s) at
various positions in the radio path are partially correlated, we require
the spatial covariance function of /3 to relate the variance of /3 to the
variance of the path attenuation a. However, raining intervals at
separated observation points are not always coincident. Hence,
definition of the spatial covariance function for /3 requires a time base
for all /3s common to all observations. A natural common time base
fulfilling this requirement is the total time, including both raining and
nonraining intervals. This means the unconditional* statistical means
and variances of and a are also needed in this formulation. We there-
fore define

K = E u {(3(t)\ (48)

and

4u = Eu\P 2 (t)} -h (49)

as the unconditional mean and variance, respectively, of /3, where
E u {~\ denotes the (unconditional) statistical average including both
raining and nonraining time. We assume that, on a long-term basis,
/3„ and o% u are independent of position in or near the radio path of

* Fading caused by other atmospheric effects, such as multipath interference and
"earth-bulge," is not treated in this paper. Therefore, the path rain attenuation a
and the attenuation gradient /3 are taken to be identically zero during nonraining
time.

1074 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

interest; therefore, we omit the position argument (s) in eqs. (48) and
(49). Similarly,

a„ = ^W/)| (50)

d u = E u \a>(t)} -SlI (51)

are the unconditional mean and variance, respectively, of a.

Based upon the definitions (48) to (51) and the relationships (22)
to (29), it can be shown that conditional and unconditional means
and variances are related by

B = /3„/P„(0) (52)

a = a u /P (L) (53)

Sj = In
and

Po(0)[l+^f]J (54)

S a 2 =ln |Po(L)[l+?|]|- (55)

Obtaining the unconditional statistical averages of both sides of eq.
(6) yields

a u = PuL. (56)

Substituting eqs. (6) and (56) into definition (51) yields

X f [(3(s,t)-l3(s',t)-dvdv'\ -&L*

Jq(z') J J

= Jo W) •/<?<*> j h 'W)

X f [{EMS, i)-/3(s', tn ~ fil) -dvdv'. (57)

Let us define a spatial covariance function 34 - 35 C^(s, s') for /3(s, and
/3(s', t) such that

C,(s, s') = E u lf3(s, t)-p(s', t)} - f u . (58)

In other words,

* u (s, s') = ^%^ (59)

is the (spatial) correlation coefficient 34 ' 35 between /3(s, t) and /3(s', t).
Substituting definitions (58) and (59) into (57) yields

«L= 4n-L 2 -H(L), (60)

RAIN ATTENUATION 1075

where

H ^ -bf*m L ff'm L, f^ s ' )dvdv '- (61)

Substituting (56) and (60) into (55) gives

SUL) = lnP (L)
Combining (54) and (62) gives

S*(L) = lnP (L)-(l + H(L)

l + ff(L)-4"l- (62)

R J

exp (Sj) _ x

(63)

Po(0)

Combining eqs. (24), (25), (52), (53), and (56) gives

fc (13. fc .L.j^.«p[az5]. (64)

This completes the derivation for S£(L) and a m (L).

APPENDIX B

Lognormal Distribution of Point Rain Rate

Figures 10 to 13 display the distributions of 2-s point rain rate
observed in Miami, Florida; Urbana, Illinois; Atlanta and Palmetto,
Georgia; Merrimack Valley, Massachusetts; Holmdel, New Jersey;
and Southern England. The time bases range from 1 to 6 years. It can
be seen that these distributions of 2-s point rain rate are very close to
the lognormal approximation in the range below 100 mm/h. The rain
rates beyond 100 mm/h are generally separated by more than 3 sigma
from the median, and constitute the tail of the lognormal distribution.
A very long observation time (e.g., more than 20 years) is necessary
to obtain stable statistics of extreme rain rates beyond 100 mm/h. 45-48
Since the time bases of the data in Figs. 10 to 13 are much less than
20 years, the deviations of the data from the lognormal distributions
in the tails are not unexpected.

The rain gauge integration time T in the original data range from
1.5 s to 2 min, depending upon the source. As discussed in Section 2.3,
the appropriate integration time T, corresponding to 1-m 3 sampling
volume in our formulation, is about 2 s. From 2-year experimental
data at Holmdel, New Jersey, Bodtmann and Ruthroff 15 have obtained
an empirical relationship for the dependence of point rain rate dis-
tributions on rain gauge integration t time in the range

1.5 s ^ T ^ 120 s. (65)

This empirical result enables us to convert the original data into the
2-s point rain rate distributions shown in Figs. 10 to 13.

1076 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

10"

RAIN GAUGE INTEGRATION TIME 1=2 SECONDS
O 3 YEARS (5/69-4/72)
• 3 YEARS (1971-1973)
A 4 YEARS (5/61-5/62.63.70,71)
LOGNORMAL APPROXIMATION

O URBANA.ILL.

SOUTHERN MERRIMACK VALLEY. MASS.

ENGLAND

40

80 120 160

RAIN RATE IN mm/h

240

Fig. 10 — Lognormal distribution of 2-s point rain rate at Urbana, Illinois ; Merri-
mack Valley, Massachusetts ; and Southern England.

APPENDIX C

Derivation tor Path Length Dependence of P„(L)

Let Po(Li) and P (Li + AL) be the probabilities that rain falls on
the radio path with length Li and an extended path of length Li + AL,
where AL is a small incremental length. The relation between P (I»i)

RAIN ATTENUATION 1077

80 1 20 1 60

RAIN RATE IN mm/h

240

Fig. 11 — Lognormal distribution of 2-s point rain rate at Miami, Florida and
Holmdel, New Jersey.

and Po(Li + AL) can be written as

Po(L, + AL) = Po(Li) + APo(AL), (66)

where APo(AL) is the incremental probability of rainfall associated
with the incremental length AL. This incremental probability can be
written as

APo(AL) = AP (AL|no rain for ^ L ^ L x )

•P(no rain for g L ^ U), (67)

where AP (AL|no rain for ^ L ^ Li) is the (incremental) prob-
ability that rain falls on the incremental length AL under the condi-

1078 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

1(T

< 10- 2

RAIN GAUGE INTEGRATION TIME T=2 SECONDS

O 2 YEARS (11/70-10/71
8/73-7/74)
• 6 YEARS (1966-1970.1973)

LOGNORMAL APPROXIMATION

PALMETTO.GA.
ATLANTA. GA.

B -«.

40

80 120 160

RAIN RATE IN mm/h

Fig. 12 — Lognormal distribution of 2-s point rain rate at Palmetto and Atlanta,
Georgia.

tion that rain is not falling on the path L\. This condition is required
because rainfall on L\ and AL are partially correlated.
We assume that

AP„(AL|no rain for g L £ L x ) « AL. (68)

The justification for this assumption is

(i) P (L) is expected to be a smooth, continuous function of L, i.e.,

lim P„(Li + AL) = Po(L x ),

Al,-»0

lim AP (AL|no rain for ^ L g Li) = 0.

AL-0

(69)
(70)

RAIN ATTENUATION 1079

<

—
O

£ 10- 1

10- 3 -

RAIN GAUGE INTEGRATION TIME T = 2 SECONDS
O 1 YEAR (8/73-7/74)
LOGNORMAL APPROXIMATION

PALMETTO, GA.

40

80 120 160

RAIN RATE IN mm/h

200

240

Fig. 13 — Lognormal distribution of 2-s point rain rate at Palmetto, Georgia.

(u) The rain-gauge network data (Fig. 3) indicate that the slope of
Po(L) is not zero, i.e.,

APo

AL

^ 0, for L^O.

Let b be the proportional parameter in assumption (68). Then

AP (AL|no rain for ^ L ^ U) = b{L)-AL. (71)

The unknown proportional parameter b(L) will be determined from
rain-gauge network data.
By definition,

P(no rain for ^ L ^ L x ) = 1 - P (Li).
Combining eqs. (67), (71), and (72) yields

APo(AL) - 6(L)-AL-[1 - Po(L,)],

(72)
(73)

1080 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

from which

dP {L)/dL
1 - Po(L)

Integrating (74) yields

= b(L). (74)

P (L) = 1 - c exp I - J b(z)dz

(75)

where c is an unknown constant to be determined by the condition

limPo(L) = Po(0). (76)

L-H)

Applying condition (76) to (75) gives

P (L) = 1 - [1 - Po(0)] exp I - j* b(z)dz\ ■ (77)

Since the rain-gauge network data yield P (L) at only a few dis-
crete distances, we need the quantized version

AP,(y/AL a h(Li)f i=h2 ,S,.--,n, (78)

of eq. (74) for estimating b(L). By using eq. (78) and the rain-gauge
network data in Fig. 3, we can calculate 6(L,) at several discrete
points. From these results, we find that b(L) can be approximately
described by the empirical formula

^Sm" (79)

Substituting (79) into (77) and carrying out the integration yields

1 - Po(0)

which is the same as (20).

Figure 3 shows that the empirical result (80) is reasonably close to
all the data points measured by the Florida rain-gauge network.
Admittedly, eqs. (79) and (80) are empirical. Further theoretical work
and multiple rain-gauge experiments are needed to improve these
approximations.

APPENDIX D

Transmission Loss Due to Wet Radomes

A 20-GHz experiment by Anderson 49 on a section of a 90-ft diam-
eter radome, pertaining to an earth satellite radio link, indicated a
transmission loss of 2 to 3 dB at 10 mm/h rain rate when the radome

RAIN ATTENUATION 1081

P.(£)Sl- , l ~fAu - ( 8 °)

was new. However, after 6 months of weathering, the transmission
loss increased to 8 dB at 10 mm/h rain rate. A 4-GHz experiment 50
on an earth-satellite radio link indicated a 3-dB transmission loss re-
sulting from the wet radome. These experimental data agree rea-
sonably well with theoretical calculations 51-64 for rain rates ^ 10 mm/h,
assuming laminar water flow on hemispherical radome surface pointing
towards the zenith.

For typical 11- and 18-GHz terrestrial radio paths, the transmission
losses from wet radomes are expected to be smaller than those of earth-
satellite radio links because of the smaller radome size, different
radome shape, and orientation. However, the terrestrial radio passes
through a pair of radomes on each link ; therefore, the contribution of
wet radome loss to the total path attenuation may not be negligible.
Theoretical calculation of wet radome attenuation pertaining to
terrestrial radio links is not available at the present time because of
the difficulty in calculating the nonuniform thickness of the water film.
A semiquantitative experiment 55 was carried out on the 12.2-GHz radio
link between Murray Hill and Crawford Hill, New Jersey (22 miles).
The 10-ft dish antenna was covered by a cone-shaped radome that
was made of resin-coated fiberglass and, at 10 years of age, was well
weathered. Water was sprayed on the radome-covered antenna by a
manually controlled sprinkler. The results indicated that a uniform
light sprinkle caused approximately 2.5-dB attenuation, whereas a
very heavy spray (maximum stream of water) caused between 4- and
7-dB attenuation. After the spray was turned off, 2 to 3 minutes
elapsed before the signal recovered to within 1 dB of its nonfaded level.
The residual wet radome attenuation is estimated to be 0.5 dB.

On an 18-GHz, 4.3-km path at Merrimack Valley, Massachusetts,
Kenny 36 has also observed a residual wet radome attenuation of 0.75
dB (i.e., 1.5 dB for two radomes).

APPENDIX E

List ot Symbols and Their Definitions

A„ Collecting aperture of a rain gauge.

A r Attenuation by two wet radomes on a radio link.

C/j(s, s') Spatial covariance function of /3(s, t) and /3(s', t) as de-
fined by eq. (58).

dv = pdpd<f>dz.

El\~\ Conditional statistical (time) average under the condition

that the point rain rates, in the radio path of length L
(see Fig. 1), are not all zero.

1082 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1975

E {~\ Conditional statistical (time) average under the condition

that the point rain rate (defined in Section 2.3) is not
zero at the position of interest.

E u {~) Unconditional statistical (time) average including both

raining and nonraining time.

erfc(~) Complementary error function.

F Q Fade margin of radio links.

G Characteristic distance defined in eq. (35). See also Sec-

tion 4.1.

h(z) Radius of the circular cross section of radio beam at a

distance z from the transmitter. See Fig. 1 and eq. (3).

H(L) Defined by eqs. (32) and (36).

H-Pol Horizontal polarization.

L The path length of a radio link; see Fig. 1.

L A reference repeater spacing defined by eq. (47).

In (~) Natural logarithm.

P(a ^ A) Probability that rain attenuation a exceeds A.

P(R ^ r) Probability that rain rate R exceeds r.

P(0 ^ B) Probability that attenuation gradient /3 exceeds B.

Po(L) Probability that rain is falling on a radio link of length L.

P (0) = linu-.oPo(£) ; the probability that rain is falling at the

position of interest.

Q (z) = wh 2 (z) ; area of the circular cross section of radio beam

at a distance z from the transmitter. See eqs. (3) and (4).

R(s, t) Point rain rate measured by a 1-m 3 sampling volume

located at s.

R m Median value of the point rain rate R during raining time.

s A vector to denote the position (p, <f>, z).

S a Standard deviation of In a during raining time.

Sr Standard deviation of In R during raining time.

St Standard deviation of In /S during raining time.

T Integration time of rain gauge.

t Time.

Tcv Defined in eq. (10).

F-Pol Vertical polarization.

V r Average falling velocity of raindrops.

AV Incremental sampling volume for measurement of point

rain rate. See Section 2.3.

z Distance from the radio transmitter (Fig. 1).

a Rain attenuation in decibels.

a m Median value of a during raining time.

a Mean value of a during raining time.

RAIN ATTENUATION 1083

a u Unconditional statistical mean of a as defined by eq. (50).

Point rain attenuation gradient measured in dB/km by a

1-m 3 sampling volume as discussed in Section 2.3.

m Median value of /3 during raining time.

/3 Mean value of /3 during raining time.

/3,(z, t) Average of /3(s, t) over the circular cross section Q of the

radio beam. See eqs. (5) and (7).

/3 U Unconditional statistical mean of /3 as defined by eq. (48).

d = 4> - <f>'.

7 A parameter defined by eq. (12) relating point rain atten-

uation gradient and point rain rate R.

X Radio wavelength.

rj A parameter defined by eq. (12) relating point rain atten-

uation gradient /3 and point rain rate R.

a a Standard deviation of a during raining time.

a au Unconditional standard deviation of a as defined by

eq. (51).

<t/3 Standard deviation of /3 during raining time.

<T0 U Unconditional standard deviation of /3 as defined by

eq. (49).

p Radial distance from the z-axis in the cylindrical co-

ordinate system in Fig. 1.

<i> Angle in the cylindrical coordinate system in Fig. 1.

^ u (s, s') Correlation coefficient between /3(s, t) and /3(s', t) as
defined by eqs. (33) and (59).

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